As argued in the proof of Theorem REF , if the spectral radius \({\rho }_G\)  is bounded,
then the same holds for the maximum degree \({\Delta }\)  as we always have that \({\Delta }\le ({\rho }_G)^2\le {\Delta }^2\) .
Since both \({\rho }_G\)  and \({\Delta }\)  are bounded, for fugacity \(0\le \lambda \le (1-\epsilon ){\lambda _c}({\rho }_G)\) ,
Theorem REF  implies that \({\rho }\left( {{\mathcal {I}}}^{\Lambda ,\tau }_{G} \right)=O(1)\) 
for any \(\Lambda \subseteq V\)  and \(\tau \in \lbrace \pm 1\rbrace ^{\Lambda }\) .
Then, Theorem REF  follows by a standard application of Theorem 1.9 in [1]}.

